Treffer: A quasi-theta product in Ramanujan's lost notebook

From the paper's abstract and introduction: On page 209 of his ``Lost Notebook'', Ramanujan records an unusual product formula, reminiscent of a product of theta functions. In this paper, we prove the formula, offer some generalizations, and indicate some further connections with Ramanujan's work. Here is the formula: \[ \begin{multlined} \left\{ \prod^\infty_{n=0} \left(\frac{1-(-1)^n q^{(2n+1)/2}} {1+(-1)^n q^{(2n+1)/2}} \right)^{2n+1} \right\}^{\log q}\left\{ \prod^\infty_{n =1} \left( \frac {1+(-1)^n iq^{\prime n}}{1-(-1)^niq^{\prime n}} \right)^n\right\}^{2\pi i}= \\ =\exp\left(\frac {\pi^2}{4}- \frac{k_3F_2 (1,1,1; \tfrac 32,\tfrac 32;k^2)} {_2F_1(\tfrac 12,\tfrac 12;1;k^2)} \right), \end{multlined} \] where \(q=\exp(-\pi K'/K),\) \(q'= \exp(-\pi K/K')\) and \(0